Coarse cone quotients

TL;DR

This paper investigates the coarse motive of quotient spaces formed from cones of uniform bornological coarse spaces with group actions, revealing conditions where the coarse assembly map fails to be an equivalence.

Contribution

It extends coarse homotopy theory to analyze quotient cones, demonstrating non-equivalence of the coarse assembly map under certain ergodic measures.

Findings

Coarse assembly map is not an equivalence for certain quotient cones.

Uses formalism of coarse homotopy theory based on bornological coarse spaces.

Adapts ideas from recent research to new setting.

Abstract

We study the coarse motive of the quotient $\mathcal{O}^{\infty}(X)//G$ of the cone of a uniform bornological coarse space $X$ with $G$-action. If $X$ admits a sufficiently ergodic probability measure, then we show that the coarse assembly map for $\mathcal{O}^{\infty}(X)//G$ is not an equivalence. The main ideas are taken from a recent paper by C. Kitsios, T. Schick and F. Vigolo (arXiv:2504.21811) and adapted to the formalism of coarse homotopy theory based on bornological coarse spaces developed by A. Engel and the author.